The visitors saw Ryan Giggs smash the crossbar from range while Adnan Januzaj had a goal ruled out for offside in the first half, but it was Gus Poyet's men who took the lead in stoppage time of the first period as United's veteran Welshman bundled the ball into his own net.
The Red Devils responded immediately after the restart, Nemanja Vidic levelling the scores from a corner, but when Tom Cleverley felled substitute Adam Johnson inside the box, Liverpool loanee Borini was able to restore the hosts' lead from the penalty spot.
|By Jamie Dunn
The Capital One Cup may not be high on Manchester United's list of priorities this season, but after defeat to Tottenham and an FA Cup exit to Swansea City, the overall picture is not a good one for David Moyes.
It was a poor game in truth, in which only Adnan Januzaj really impressed for the Old Trafford outfit, but they were back in the tie in the second half thanks to a Nemanja Vidic corner, after Ryan Giggs had bundled the ball into his own net before the interval.
And United might have had at least a draw if match official Andre Marriner hadn't given a penalty for a soft challenge by Tom Cleverley on Adam Johnson, who must have felt only minimal contact.
The tie is far from over with the second leg at Old Trafford, but this defeat makes the second visit of Swansea in the space of seven days all the more important.
The hosts enjoyed the majority of possession in the opening exchanges and had the first glimpse of goal on 12 minutes with Borini's scuffed effort.
United began to settle into the match with Januzaj looking particularly threatening, and Moyes’ side went agonisingly close to breaking the deadlock when Giggs’ deflected strike rattled the crossbar after 25 minutes.
The visitors also thought their pressure had told on 38 minutes when Januzaj fired home after initially striking against Giggs, but the linesman ruled the latter was offside when he blocked the ball.
And United’s failure to take their chances counted against them on the stroke of half-time as Wes Brown knocked Sebastian Larsson's free kick across the six-yard box where Giggs, sliding in to challenge former United team-mate Phil Bardsley, inadvertently applied the final touch from close range.
It did not take long for United to restore parity after the break, however, as Vidic leapt highest to head home Cleverley's whipped corner on 51 minutes.
But, rather than building on their equaliser, it was Sunderland who came closest to regaining the lead on 62 minutes when Larsson's volley was parried to safety by David de Gea.
Three minutes later, Poyet's side did move ahead when Johnson - who had replaced Emanuele Giaccherini from the bench - was the victim of a clumsy tackle by Cleverley in the area and Borini blasted the resulting penalty into the roof of the net.
With 20 minutes remaining, the lively Januzaj almost levelled with a bending effort that curled wide of Vito Mannone's far post.
United piled on the pressure in the closing stages with Januzaj twice coming close and an effort from Evra just evading the far post, but Sunderland withstood the barrage to ensure they will take a lead to Old Trafford in the second leg on January 22.